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Longitudinal waves and transverse waves are all around us. The light that helps us see everything around us exhibits wave-like properties; throw a stone in a pond, and you will see waves expanding in all directions from the point of impact; radio waves, microwaves, X-rays, sound waves, tsunamis – energy from different natural and man-made phenomena emanate in the form of waves.
But what are they exactly? Are all waves the same? If not, then how do they differ? What are the different kinds of waves in existence, and what are their categories?
This write-up investigates the above questions and then focuses on analyzing the differences by comparing transverse waves vs longitudinal waves.
Well, waves are rhythmic disturbances or vibrations that propagate through a medium and transfer energy along with them without transferring matter. As we pointed out, waves are of different types, and they are all around us, from all the light and sound to the radio waves emanating from smartphones and the huge ocean breakers people surf on.
If we apply elements of deductive reasoning to wave motions evident all around us, we will note some key things à
Waves are energy transmission processes that can be categorized as per the nature of their motion, the energy they transfer, and their need for any propagation medium.
A primary categorization of waves is based on their need for a medium of propagation. Matter and electromagnetic waves are the two primary wave categories via which energy is transferred in the world around us.
Vibrations cause the molecules of the medium to collide against one another, and these collisions gradually transfer energy from one point of a medium to another.
The oscillating electromagnetic fields have different components in different directions and different phases. As you may know, variations in an electric field will induce a magnetic field in the vicinity & vice versa. Electromagnetic waves, thus, are oscillating electromagnetic fields in motion.
Where does all the energy come from? It combines all the energy stored in the mutually orthogonal electric and magnetic fields. There exists a specific energy density within electromagnetic waves and it is the sum of the energy density contained within the mutually perpendicular electric [Ey(x,t) = E0 cos(kx – ωt)] and magnetic fields [Bx (x, t) = B0 cos(kx – ωt)] and is given as à
u(x,t) = 0.5* ε*0 E2 + 0.5 * (1/m0) * B2
Another major categorization of waves derives from the differences in motion of different kinds of waves. Classified into transverse and longitudinal, these two categories are the subject of discussion in this article.
Crests/peaks and troughs/valleys develop in the medium as it experiences periodic displacement at right angles to the direction of the waves; propagation. Ripples on water surface & their deadlier versions, tsunamis, secondary(S) or shearing seismic waves, and electromagnetic waves exhibit transverse wave motion.
Given the periodic nature of wave motion, longitudinal waves propagate and transfer energy by compressing and rarefying the propagation medium. Sound and ultrasound waves, vibrations in spring, and primary(P) or compressional seismic waves propagate using longitudinal wave motion.
Now, let’s look at the two wave motion variations.
As we mentioned, transverse waves are sets of all those waves that propagate energy by oscillating the medium particles orthogonally to the direction of wave motion. One of the most powerful examples of transverse waves is the secondary (S) or shearing seismic waves produced during earthquakes that cause rock particles to slide up & down and produce immense amounts of shearing stress.
Other prominent examples of transverse wave motion are à
The mutually orthogonal oscillation of medium particles concerning the wave motion produces sinusoidal waveforms within the medium. As we all know, the periodic nature of waves lends certain harmonics properties to the phenomenon that are depicted through certain key parameters, namely, amplitude, frequency, and wavelength.
Frequency and wavelength together determine the characteristics of any kind of wave, irrespective of the motion or wave type, and the same goes for transverse waves.
For transverse waves, the direction of the wave’s displacement is linearly polarised, while the magnitude of the displacement is sinusoidal.
The above is true for any progressive wave, transverse or longitudinal. For a transverse sinusoidal wave, the equation is given as
Yy(x,t) = a sin (kx -ωt + F)
Where Fdepicts an argument for the sine function that turns the equation into a linear combination of sine and cosine functions, a is the wave amplitude, k is the wave number or the inverse of the wavelength, and ω is the angular or harmonic velocity.
The above wave equation depicts how medium particles displace and oscillate. Differentiating the above equation gives us the PARTICLE VELOCITY of transverse waves.
Both transverse and longitudinal waves are traveling waves, and both can be depicted using sinusoidal functions if the waveform is sinusoidal. The nature of waveforms, however, depends upon the nature of the energy and the medium.
Like any harmonic or oscillating motion, wave motion involves restoring force and stabilising the medium particles once the wave energy has moved on. If we consider transverse waves travelling through a string, then two forces contribute to the wave’s motion through the string, à, the tension of the string T (the restoring force) and the mass density of the string m (the inertial property of the string).
The wave velocity, in terms of these two forces, is then given as
V= Ö(T/ m)
Note how the velocity of a transverse mechanical wave does not depend on the waveform, the frequency, or the amplitude. This is the WAVE VELOCITY of transverse and is independent of time.
However, high-velocity waves do depend substantially on their frequencies.
Now it is time to look at the nature and features of longitudinal waves.
As mentioned, longitudinal waves traverse by compressing and rarefying the propagation medium. The medium particles oscillate here, too, as expected from periodic motions; only in this case, the oscillations are in the direction of the propagation of the wave. However, as in the case of transverse waves, only the energy propagates from one point to another, not the medium.
Some of the most prominent examples of longitudinal waves in real life are:
Sinusoids are considered the purest of all waveforms, and sinusoidal longitudinal waves can be depicted with the following equation à
Yx(x,t) = a sin (kx -ωt + F)
The parameters and variables remain the same. As the medium particles vibrate or oscillate parallelly along the wave, the y(x,t) depicts the horizontal motion of the medium particles.
Differentiating the above equation gives us the particle velocity of longitudinal waves à
Vx (x, t) = ∂y(x,t)/∂t=−Aωcos(kx−ωt+φ).
If we consider the speed of sound through a solid medium, the compressions and rarefactions depend on the mass and elastic properties. The compressional stress and strain are determined by the Bulk modulus of the medium and are given as à
B = -[ΔP/(ΔV/V)]
ΔP is the change in pressure that creates all the compressive stress, while ΔV/V is the volumetric strain generated due to the stress.
If we consider compressive stress to be the vital force, then the mass density of the medium is the other key factor for longitudinal mechanical waves. Bulk modulus has the same dimensions as restorative stress, and thus, the longitudinal wave velocity can be calculated as à
Where ρ is the mass density of the medium and provides the countering inertial property.
Now that we have mulled over the key features, properties, and behaviour of both transverse and longitudinal waves, let’s pit one against the other.
Apart from the direction and nature of displacement of medium particles, there are not many differences between transverse & longitudinal waves. If the underlying waveform is sinusoidal, the wave equations are the same in both cases, with only the nature of displacements being different.
Here are the biggest differences between transverse and longitudinal waves.
Transverse Waves à The direction of wave motion is perpendicular or orthogonal to the direction of particle oscillations.
Longitudinal Wavs à The direction of wave propagation is parallel or in the same direction as that of medium particle oscillations.
Transverse Waves à Particle displacements are perpendicular to the direction of wave motion.
Longitudinal Waves à Particles oscillate in the direction of wave propagation through compressions and rarefactions by changing the mass density of the medium particle.
Both transverse and longitudinal waveforms can be sinusoidal in nature, as we saw in the sections above. The exact size and shape of any waveform depend on the source of the energy or disturbance and the nature of the waveform.
Oscillations of medium particles depend on the type of wave motion. Transverse wave motions make medium particles oscillate orthogonally to the direction of wave propagation. On the other hand, longitudinal waves cause medium particles to vibrate or displace in the direction of the wave.
The energy and power transferred by a wave do NOT depend upon the nature of the wave motion. Amplitude and frequency are the factors that determine the energy contained in a wave. Wave equations contain both these parameters and depict discrete packets of energy.
Waves with high amplitudes cause the medium particles to vibrate at higher magnitudes. If the wave frequency is high, the medium undergoes massive vibrations more frequently. The amount of energy transferred through the medium is thus higher if amplitudes and frequencies are higher.
The total mechanical energy of any wave is the total of its kinetic and potential energies.
ΔK=0.5* (μΔx) vy2
where μ is the mass density, Δx is the mass element, ΔK is the kinetic energy of that element, and vy is the mass velocity. The wave traversing through the string is undergoing transverse motion.
For longitudinal wave motion, only the mass or particle velocity direction changes.
dK = (0.5) * (μdx)* A2ω2cos2(kx)
If we integrate across the entire length of the string, the total kinetic energy due to the wave comes to à
Kl = (1/4) * A2*ω2* μ*l
Where lis the length of the entire string.
Ul = (1/4) * A2*ω2* μ*l
(Can you find out how and why the kinetic and potential energy of the string have the same expression?)
El-= (0.5) * A2*ω2* μ*l
The time-average power of a sinusoidal transverse mechanical wave can be calculated by taking a specific reference point and then dividing the equation above by the time taken by the wave to cross that specific point.
Let’s wrap up things up with a quick reiteration of everything discussed.
The oscillations can be determined using a specific wave equation. If the waveform of those oscillations is sinusoidal, then the waveform equation is given as
Yx(x,t) = a sin (kx -ωt + F)
Energy propagates through differential compressive pressure across the body of the medium. The amplitude of the above equation depicts the distance between the medium particles during compression. The closer they are, the higher the amplitude of the waveform.
Transverse waves have crests & troughs that are the counterparts of compressions and rarefactions in longitudinal waves.
If studying waves and motion seems like a struggle, you need to put in more effort. Solid advanced mathematical concepts and a clear idea of mechanics are essential to understand and master every aspect of waves and wave motion.
Put in the effort, and go through definitions & examples from different sources. solve as many sums as possible. If things still seem problematic, connect with the subject matter experts of MyAssignmenthelp.com.